On the Accuracy and Stability of Various DG Formulations for Diffusion
For computational scientists using DG methods for diffusion problems, this work provides a comparative analysis of stability and accuracy trade-offs among popular formulations, though the results are incremental and specific to uniform grids.
This paper analyzes the stability and accuracy of several discontinuous Galerkin (DG) viscous flux formulations (SIPG, BR1, BR2, LDG) using von Neumann stability analysis and combined-mode analysis. Key findings include that LDG provides larger dissipation for high-wavenumber components and lower error bounds for long-time simulations, while BR1 with added dissipation can match BR2's stability.
In this paper, we study the stability (in terms of the maximum time step) and accuracy (in terms of the wavenumber-diffusion properties) for several popular discontinuous Galerkin (DG) viscous flux formulations. The considered methods include the symmetric interior penalty formulation (SIPG), the first and second approaches of Bassi and Rebay (BR1, BR2), and the local discontinuous Galerkin method (LDG). For the purpose of stability, we consider the von Neumann stability analysis method for uniform grids with a periodic boundary condition. In addition, the combined-mode analysis approach previously introduced for the wave equation is utilized to analyze the dissipative error. This new approach can be used to study the performance of a particular DG and Runge-Kutta DG (RKDG) scheme for the entire extended wavenumber range. Thus, more insights into the robustness as well as accuracy and efficiency can be obtained. For instance, the LDG method provides larger dissipation for high-wavenumber components than the BR1 and BR2 approaches for short time simulations in addition to a lower error bound for long time simulations. The BR1 approach with added dissipation can have desirable properties and stability similar to BR2. For BR2, the penalty parameter can be adjusted to enhance the performance of the scheme. The results are verified through canonical numerical tests.